On the Resolution of Time Problem in Quantum Gravity Induced from Unconstrained Membranes

TL;DR

This paper develops a relativistic unconstrained membrane model for quantum gravity, providing a solution to the problem of time by localizing wave functionals in evolving space-time regions, and discusses quantization and sources.

Contribution

It introduces a novel unconstrained membrane framework for quantum gravity that resolves the problem of time through localized wave functionals in evolving space-time regions.

Findings

Wave functional solutions are localized in a moving space-time region.

The approach resolves the problem of time in quantum gravity.

Inclusion of sources avoids issues with Maxwell fields for charged particles.

Abstract

The relativistic theory of unconstrained $p$-dimensional membranes ($p$-branes) is further developed and then applied to the embedding model of induced gravity. Space-time is considered as a 4-dimensional unconstrained membrane evolving in an $N$-dimensional embedding space. The parameter of evolution or the evolution time $\tau$ is a distinct concept from the coordinate time $t = x^0$. Quantization of the theory is also discussed. A covariant functional Schr\" odinger equations has a solution for the wave functional such that it is sharply localized in a certain subspace $P$ of space-time, and much less sharply localized (though still localized) outside $P$. With the passage of evolution the region $P$ moves forward in space-time. Such a solution we interpret as incorporating two seemingly contradictory observations: (i) experiments clearly indicate that space-time is a continuum in which events are existing; (ii) not the whole 4-dimensional space-time, but only a 3-dimensional section which moves forward in time is accessible to our immediate experience. The notorious problem of time is thus resolved in our approach to quantum gravity. Finally we include sources into our unconstrained embedding model. Possible sources are unconstrained worldlines which are free from the well known problem concerning the Maxwell fields generated by charged unconstrained point particles.